Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated nonperturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme provides a good description of the spectral line shape at zero and finite temperatures both in the weak- and strong-coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling.
We use the numerical renormalization group method to study an Anderson impurity in a conduction band with the density of states varying as with r > 0. We find two different fixed points: a local-moment fixed point with the impurity effectively decoupled from the band and a strong-coupling fixed point with a partially screened impurity spin. The specific heat and the spin susceptibility show power-law behaviour with different exponents in the strong-coupling and local-moment regimes. We also calculate the impurity spectral function which diverges (vanishes) with in the strong-coupling (local-moment) regime.
The impurity Green function and dynamical susceptibilities for the two-channel Anderson impurity model are calculated. An exact expression for the self-energy of the impurity Green function as ratio of two correlation functions is given. The imaginary part of the self-energy scales as $\sqrt{\ensuremath{\mid}\ensuremath{\omega}∕{T}_{K}\ensuremath{\mid}}$ for $T\ensuremath{\rightarrow}0$, serving as a hallmark for non-Fermi behavior. The many-body resonance is pinned to a universal value of $1∕(2\ensuremath{\pi}\ensuremath{\Delta})$ at $\ensuremath{\omega}=0$ for arbitrary local occupation in contrary to the single-channel Anderson impurity model. Its shape becomes increasingly more symmetric for the Kondo regimes of the model. The dynamical spin and channel susceptibilities are governed by two energy scales
TK and
Th, and their ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ approach a constant value for $\ensuremath{\omega}\ensuremath{\rightarrow}0$.
We investigate several definitions of the time-dependent spectral function $A(\ensuremath{\omega},t)$ of the Anderson impurity model following a quench and within the time-dependent numerical renormalization group (TDNRG) method. In terms of the single-particle two-time retarded Green function
Gr(t1,t2), the definitions we consider differ in the choice of the time variable
t with respect to
t1 and/or
t2 (which we refer to as the time reference). In a previous study [H. T. M. Nghiem et al., Phys. Rev. Lett. 119, 156601 (2017)], we investigated the spectral function $A(\ensuremath{\omega},t)$, obtained from the Fourier transform of
Im[Gr(t1,t2)] with respect to the time difference ${t}^{\ensuremath{'}}={t}_{1}\ensuremath{-}{t}_{2}$, with time reference
t=t2. Here, we complement this work by deriving expressions for the retarded Green function for the choices
t=t1 and the average, or Wigner, time
t=(t1+t2)/2, within the TDNRG approach. We compare and contrast the resulting $A(\ensuremath{\omega},t)$ for the different choices of time reference. While the choice
t=t1 results in a spectral function with no time dependence before the quench (
t<0) (being identical to the equilibrium initial-state spectral function for
t<0), the choices
t=(t1+t2)/2 and
t=t2 exhibit nontrivial time evolution both before and after the quench. Expressions for the lesser, greater, and advanced Green functions are also derived within TDNRG for all choices of time reference. The average-time lesser Green function ${G}^{<}(\ensuremath{\omega},t)$ is particularly interesting, as it determines the time-dependent occupied density of states $N(\ensuremath{\omega},t)={G}^{<}(\ensuremath{\omega},t)/(2\ensuremath{\pi}i)$, a quantity that determines the photoemission current in the context of time-resolved pump-probe photoemission spectroscopy. We present calculations for $N(\ensuremath{\omega},t)$ for the Anderson model following a quench, and discuss the resulting time evolution of the spectral features, such as the Kondo resonance and high-energy satellite peaks. We also discuss the issue of thermalization at long times for $N(\ensuremath{\omega},t)$. Finally, we use the results for $N(\ensuremath{\omega},t)$ to calculate the time-resolved photoemission current for the Anderson model following a quench (acting as the pump) and study the different behaviors that can be observed for different resolution times of a Gaussian probe pulse.
We use the numerical renormalization group method (NRG) to investigate a single-impurity Anderson model with a coupling of the impurity to a superconducting host. Analysis of the energy flow shows that, contrary to previous belief, NRG iterations can be performed up to a large number of sites, corresponding to energy differences far below the superconducting gap Δ. This allows us to calculate the impurity spectral function A(ω) very accurately for frequencies |ω|∼Δ, and to resolve, in a certain parameter regime, sharp peaks in A(ω) close to the gap edge.
We calculate the exponents of the threshold singularities in the photoemission spectrum of a deep core hole and its x-ray absorption spectrum in the framework of a systematic many-body theory of slave bosons and pseudofermions (for the empty and occupied core level). In this representation, photoemission and x-ray absorption can be understood on the same footing; no distinction between orthogonality catastrophe and excitonic effects is necessary. We apply the conserving slave particle T-matrix approximation (CTMA), recently developed to describe both Fermi and non-Fermi liquid behavior in systems with strong local correlations, to the x-ray problem as a test case. The numerical results for both photoemission and x-ray absorption are found to be in agreement with the exact infrared power law behavior in the weak as well as in the strong coupling regions. We point out a close relation of the CTMA with the parquet equation approach of Nozi\`eres et al.